wllneg - STAT 330 - WLLN Examples If X v PAR (1, ) then X...

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STAT 330 - WLLN Examples If X v PAR (1 , θ ) then X has p.d.f. f ( x )= θ x θ +1 ,x 1 , c.d.f. F ( x ( 0 if x< 1 1 1 x θ for x 1 , inverse c.d.f. F 1 ( x )=(1 x ) 1 / θ , 0 <x< 1 , mean E ( X ( if 0 < θ 1 θ θ 1 if θ > 1 and variance Var ( X θ ( θ 1) 2 ( θ 2) . Suppose X 1 ,...,X n are i.i.d , θ ) random variables. By the WLLN ¯ X n = 1 n n X i =1 X i p E ( X θ θ 1 , if θ > 1 . If U i v UNIF(0 , 1) ,i =1 ,...,n independently then X i = F 1 ( U i U i ) 1 / θ v , θ ) i independently (see Example 2 . 4 . 3 ). If we generate , 1) observations u 1 ,...,u N using a random number generator and then let x i =(1 u i ) 1 / θ then the x i ’s are observations from the , θ ) distribution. Figure 1 is a plot of the points ( i, x i ) ,..., 500 for one simulation of 500 points for θ =5 .F igu re2 is the corresponding plot of ( n, ¯ x n ) ,n 500 where ¯ x n = 1 n P n i =1 x i for these data. We note that ¯ x n approaches E ( X )=1 . 25 as n increases. We generate a further 1000 values of x i , and plot ( i, x i ) 1500 in Figure 3 and plot ( n, ¯ x n ) , n 1500 in Figure 4. In Figure 4 we notice that
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This note was uploaded on 04/27/2010 for the course STAT 330 taught by Professor Paulasmith during the Fall '08 term at Waterloo.

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wllneg - STAT 330 - WLLN Examples If X v PAR (1, ) then X...

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